1. Introduction

Optically induced waveguides constitutes a challenging research subject whose motivation is the fabrication of complex optical circuits in 2-D or even 3-D if realized deep inside crystals. These waveguides can be formed in numerous medium such as polymers [1–3], liquid crystals [4], glass [1, 5, 6] or photorefractive crystals [7–9]. Realization of these structures would benefit from a technique allowing real time evaluation during the photo-induction process. This would provide a tool to characterize the induction process and would ultimately give an accurate control of the characteristics of the fabricated waveguides. Different techniques such as refractive index mapping using interferometry [8] or higher order mode excitation [7] are already employed for this purpose but more versatile characterization tools that take advantage of the specificity of photoinduced waveguides would be helpful.

In this paper we propose a simple method to assess the properties of photo-induced waveguides. The technique is functional providing some waveguide parameters such as index profile are known. The method relies on the spatio-temporal analysis of light distribution at the exit face of a waveguide while its index profile is gradually changing either during the induction process or when the waveguide is gradually erased. After briefly presenting the theoretical basis, the technique is applied to analyze waveguides formed by photorefractive spatial soliton-like beams in a strontium barium niobate (SBN) crystal.

2. Principle

When coherent light is launched at the entrance face of a waveguide the energy is distributed among guided and leaky modes. As the light wave propagates, it gives rise to a complex light distribution due to the coherent superposition of these multiple independent waves each traveling with a different phase velocity. This phenomenon is commonly referred as mode beating. As a result, the light distribution along propagation direction and the waveguide properties are closely interrelated. Therefore, knowledge of light distribution as it travels in the waveguide can thus lead to waveguide characteristics. However, in most case, an accurate observation of light inside the waveguide is unachievable. As an alternative we propose to analyze light distribution as a function of time at the exit face of the waveguide while the index structure gradually evolves. This techniques is especially suitable for photo-induced waveguides since measurements can be performed in real time during waveguide formation or afterwards if waveguide erasure is possible. The waveguide characteristics can be inferred accurately from the observed dynamical behaviour if some waveguide parameters are initially identified.

More specifically, we have probed the photo-induced structure with a plane-wave like beam in order to excite not only guided modes but also to launch .light in the region surrounding the waveguide which, as shown later, provides a good sensitivity and reproducibility to the technique for single mode or multimode waveguides. Interestingly, the overlap integral between the excitation beam and odd modes is null so that in general only even modes are excited. Our results are shown in a movie composed of the sequence of experimentally observed images at the exit face of the changing waveguide. This is compared with a corresponding movie constructed from numerical calculations based on a beam propagation method (BPM) using the split step Fourier technique [10].

3. Validation of the technique

To validate the technique we consider slab waveguides induced by a 1-D bright photorefractive spatial screening soliton formed in SBN:75 crystals [7, 11]. The dynamic behavior observed at the exit face of the waveguide is taken while the waveguide is decaying instead of during its formation to avoid any influence of probe beam on the writing process.

According to the photorefractive time-dependent 1-D soliton theory [12] the refractive index change produced by a photorefractive screening soliton whose profile is I(x) is given by:

Where r_eff is the effective electro-optic coefficient, E ₀ is the external applied field, I_d is the equivalent dark irradiance, T_d is the dielectric response time of the crystal in the dark and t is the photo-induction duration time. Evidently, in order to obtain the refractive index profile, the soliton intensity profile I(x) has to be self-consistently determined using the non-linear Schrödinger propagation equation [11, 12]. To form a steady-state soliton [13] the induction time has to be longer than T_dI_d/I_max while a soliton in quasi-steady-state regime [14, 15] is obtained for an induction time t ₀ = 2I _max T_d/I_d [12, 16]. Solitons in the quasi-steady-state regime have been used in this study because less stringent parameter control is needed for their creation compare to steady-state solitons. For example, background illumination is not necessary and I_d is only due to the thermal generation of free carriers. In addition because the soliton beam intensity is much higher than I_d the soliton width is intensity independent [12]. Once the soliton is formed both the applied field and the soliton beam are turned off. When subsequently probed by a large probe beam the waveguide depth is taken to decay exponentially in our model due to gradual erasure. Note that for the proposed technique precise knowledge of the waveguide decay constant is not required but on the contrary its value can be evaluated experimentally.

A 1 cm³ poled SBN:75 cubic-shaped sample is used to performed the experiments. The first step is to create a 1-D photorefractive soliton using ordinary light at 514 nm. The experiment consists in focusing the beam from a laser using a cylindrical lens into a 15 μm FWHM wide stripe at the entrance face of the SBN crystal. The stripe propagates perpendicular to the crystal c-axis while the external dc electric field E ₀ is applied along the c-axis. When the applied electric field is set to 3kV/cm the initial beam diffraction is gradually compensated as the photorefractive effect form a soliton. At the light intensity used for the experiment, the time t₀ to form a quasi-steady-state soliton is about 20 min. Then the soliton forming beam and the dc applied field are both turned off and an ordinary polarized 300 μm diameter probe beam is launched normal to the entrance face of the photoinduced waveguide. The image of the light distribution at the exit face is then recorded at regular interval using a CCD camera linked to an acquisition system.

 

Fig. 1. Observed light distribution at the exit face of a decaying waveguide initially formed by a quasi-steady-state soliton in the initial stage (a) and at two subsequent characteristic times (b, c). Movie of the entire process corresponding to 40min observation is attached (size : 1.35 Mb).

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Fig. 2. Calculated dynamic of light distribution at the exit face of a decaying soliton-induced in the initial stage (a) and at two subsequent characteristic times (b, c). Movie of the entire process is attached (size : 640 kB).

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PR waveguides can decay due to thermal excitation or if illuminated with a light beam at the appropriate wavelength. In our experiments, a probe beam is both used to sense the waveguide and to gradually erase it. The constant time associated with the decay is inversely proportional to the illumination as is usually the case for photorefractive effect. Accordingly, the probe intensity is adjusted so that the erasure process can easily be observed in real time and the images at the exit face can be capture at regular intervals.

The first waveguide is formed with a 20 min induction time which corresponds to a soliton in quasi-steady-state regime for our experimental parameters. Evolution of the intensity distribution at the exit face of the probed waveguide as the photo-induced structure decays is depicted in Fig. 1. At the start, the presence of a peak in intensity due to light trapping in the high index region of the medium can be clearly seen (Fig. 1(a)). As the waveguide weakens a remarkable feature is noticed. The contrast between the darkest and the brightest regions increases (Fig. 1(b)). The intensity distribution then becomes smooth as the refractive index distribution becomes flat (Fig. 1(c)). The entire process which lasts 40 min is summarized in the movie in Fig. 1. To model the observed dynamic behavior we consider that the initial index profile is given by Equ. (1) with the inducing spatial soliton being described by the experimental parameters of 15 μm FWHM, E ₀ = 3 kV/cm at t = t₀ while the only free parameter is the electro-optic coefficient. Assuming an exponential decay time constant for the waveguide erasure we obtain the calculated behavior presented in Fig. 2 when the electro-optic coefficient is set to 40 pm/V, a number determined by requiring good agreement with experiments. The initial waveguide refractive index profile width and depth is estimated to be, respectively, 18μm (FWHM) and 6 10^-5 with an accuracy better than 5%. The overall dynamics calculated numerically agrees well with the experimental observation. In particular the increased light modulation that appears after the waveguide has weakened is clearly reproduced by the simulation. The value of the electro-optic coefficient r₁₃ used to fit the experimental results is a bit lower than the one (70 Pm/V) usually reported in the literature [17] which can be attributed to poling of the SBN sample. The observed non monotonic decay of the trapped light as the waveguide decreases may seem surprising at first especially if we keep in mind that the waveguide is single mode. Even if the waveguide formed by the photorefractive soliton is not always single mode it is indeed the case for steady-state regime soliton with a ratio I_max/I_d close to 2 [7]. Since a quasi-steady-state soliton has identical characteristic its single mode character is also expected. We then cannot attribute the non monotonous light distribution observed at the exit face to coherent addition of multiple guided modes. The effect is instead due to the coherent overlap between the guided fundamental mode and light in the vicinity of the waveguide each having a different traveling speed consequently giving rise to beating. Although the overall behavior as a function of time depicted by theory agrees with the experiment, a simple exponential decay does not fit perfectly the time evolution of the observations. The primarily reason is that a varying light distribution is responsible in practice for the waveguide erasure contrary to the hypothesis of a uniform illumination. For instance, we observe that, in the initial stage, alteration of the observed light distribution shown in Fig. 1 is happening slower than predicted (Fig. 2). The non uniformity of the probe beam could be the reason of this difference between theory and experiment. A more accurate description of the time dependence of the process would thus imply to take into account the temporal variations of light distribution on the waveguide decay. This refinement is not necessary if the goal is to assess the geometrical and optical properties of the initial waveguide instead of determining the dynamic of the erasure. We would like to point out that no partial reconstruction of the waveguide is possible under the influence of the probe beam because of the absence of applied electric field, spatial charges tends to redistribute uniformly which induces the waveguide gradual disappearance.

 

Fig. 3. Calculated dynamic of light distribution at the exit face of a decaying waveguide initially created by a partially formed soliton in the initial stage (a) and at two subsequent characteristic times (b, c). Movie of the entire process is attached (size : 548 kb).

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Fig. 4. Observed light distribution at the exit face of a decaying waveguide initially created by a partially formed soliton in the initial stage (a) and at two subsequent characteristic times (b, c). Movie of the entire process corresponding to 20min observation is attached (size : 965 kb).

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The probe technique was then performed on a waveguide induced by a partially formed soliton. In this case, the process to form a soliton is repeated identically as in the previous experiment except that the induction time is limited to 5 minutes. Because of a shorter photoinduction time the soliton regime is not fully reached and index contrast of the waveguide is weaker. The beam diffraction is however strongly reduced and the induced index transverse profile can be considered fairly homogeneous along the 1cm crystal length. The refractive index profile is expected to be correctly described by Eq. (1) because of the closeness to soliton regime. When probed by a large beam the theoretical predicted dynamical observation is shown in Fig. 3 for an induction time set to t₀/4 while the other parameters are kept identical to the calculation in Fig. 2. The theoretical result fits remarkably well with the experimental observation from Fig. 4 giving a waveguide index depth of 3 10^-5. At start, we observed that the light focused in the waveguide is separated from the surrounding probe beam by a pronounced dark region. Afterward this contrast decreases monotonically as a function of time. These two observations along with an initial wider focused beam constitute the main differences with the behavior shown in Fig. 2. It also confirms that the technique can be used to recover waveguides properties even for single mode waveguides. Conversely, similar experiments performed with a probe beam tightly focused at the entrance face of the waveguide display dynamic behaviors with low reproducibility and are consequently difficult to interpret. The main reason is because unguided light distribution at the exit face of the waveguide is very sensitive to input probe characteristics such as beam waist or alignment.

 

Fig. 5. Observed light distribution at the exit face of a decaying waveguide initially created by a quasi-steady-state soliton and probed by an extraordinary polarized beam, in the initial stage (a) and at two subsequent characteristic times (b, c). Movie of the entire process corresponding to 28min observation is attached (2.2 MB).

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Finally, another soliton-induced waveguide is formed in the crystal in order to apply the technique to a multimode waveguide. The waveguide is still produced with an ordinary polarized quasi-steady-state soliton with identical experimental parameters used for the results in Fig. 1 except that we now consider the index change corresponding to an extraordinary polarized beam. For that reason, the waveguide is probed with a beam polarized along the crystal c-axis thus taking advantage of the strong r₃₃ electro-optic coefficient of SBN. The dynamics observed at the exit face is presented in Fig. 5. Dramatic changes are clearly indicated in the movie. The complex mode beating reveals that the waveguide supports 3 guided modes. Indeed, the three oscillating intensity peaks present in the central part of the induced waveguide are due to the interference between the fundamental and the third guided mode. The movie also reveals that at some instant an almost homogeneous intensity appears despite the presence of a strong waveguide as for example, shown in Fig. 5(a). This makes clear that a simple observation of light distribution at the exit face of a fixed waveguide is not sufficient to deduce its guiding properties. In the present case observation limited to the initial condition would lead to the wrong conclusion that the waveguide is very weak. On the contrary, the observed dynamics for a decaying waveguide offers an accurate way to verify the guiding characteristics. At the beginning of the process unexpected light modulations are visible on both sides of the waveguide. We believe this index perturbation located away from the waveguide is produced in the early stages of soliton formation. Actually, when the writing beam initially diffracts, light temporary present in this region gives rise to a non-uniform space charge field prior to being trapped as a soliton in the central region. Note that these perturbations are hardly seen when the probe is ordinary polarized since the associated index change is weak in that case. Moreover, we note that light has an even distribution during most part of the dynamic as expected from even mode excitation. However, toward the end of the erasure process we think that the appearance of an asymmetric light profile is due to excitation of the second mode, which has an odd profile. This could be the consequence of beam fanning [18] which can potentially tilt the trajectory of the probe beam and can thus lead to excitation of the odd second mode until the waveguide becomes single mode. Despite these two abnormal features the overall behavior observed in the vicinity of the waveguide can be reproduced by BPM calculations. The predicted dynamic behavior is shown in Fig. 6 using an electro-optic coefficient r₃₃ of 700 pm/V, while other parameters are set identical to those used for the results in Fig. 2. This again falls short of the value reported in the literature (1300 pm/V). Correlation between the two predicted electro-optic coefficients reinforces the hypothesis of a partially poled crystal. In the case of a multimode waveguide, use of a large probe beam provides a way to excite all even guided modes supported by the waveguide. As the waveguide decays, it consequently exhibits a complex but reproducible dynamic behavior which unambiguously reveals the main characteristics of the waveguides.

 

Fig. 6. Calculated dynamic of light distribution at the exit face of a decaying waveguide initially created by quasi-steady-state soliton and probe by a extraordinary polarized beam in the initial stage (a) and at two subsequent characteristic times (b, c). Movie of the entire process is attached (1.3 MB).

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4. Conclusion

We have proposed and demonstrated a simple technique to probe the guiding properties of photoinduced waveguides, such as, soliton induced waveguides. The technique is based on the observation of the time dependent light distribution observed at the exit face of a waveguide excited by a plane wave when its index profile is gradually evolving. Experimental demonstrations have been made on single mode and multimode 1-D soliton induced waveguides in SBN crystals. The technique can be extended to test 2-D waveguides and realtime probing can be used to determine the number of guided modes, the refractive index profile or the decay and formation time as long as some waveguide parameters are initially known.